Regional Housing Market Dynamics in Australia
A PCA-Based Approach to Factor Models
Yiran Yao
2025-08-04
Introduction
Traditional approaches to analysing housing indexes, such as hedonic or repeat-sales models, can introduce selection bias or violate underlying assumptions. Moreover, they fail to capture key regional, social, or macroeconomic factors that play a significant role in driving housing price movements.
In the recent research Sijp et al. (2025) propose a new approach that applies Principal Component Analysis (PCA) to regional SA4 level housing indexes, enabling the extraction of key drivers of price movements and offering a deeper understanding of the forces shaping housing markets.
The PCA results show that the first three principal components capture much of the behavior of local price indexes, as a linear combination of their time series explains most of the variance in the housing index. Nevertheless, the PCA-based linear model lacks robustness across different time windows, and its coefficient estimates must be derived directly from the PCA procedure.
Our exploratory research extends the PCA approach by addressing these limitations through a factor linear model, which uses time series data directly, with independent variables serving as proxies for the PCA-derived principal components. We model the factor proxies as time series using forecasting techniques to obtain their projected values. The forecasted factor series are subsequently incorporated into the linear model to generate projected housing price indexes, with time-varying coefficients included to capture evolving dynamics over time.
The resulting factor model can be applied for a variety of purposes: - To understand the key drivers of housing price movements at a national and regional level. - To generate forecasts of housing price indexes based on projected factor values. - To assess the impact of macroeconomic variables on housing prices through the regression coefficients.
Methodologies and Datasets
Methodlogies
PCA
Before introducing the factor model, Principal Component Analysis (PCA) was first applied to the original housing indexes. It is therefore useful to briefly revisit PCA and how it allowed us to identify specific factors. This work was originally conducted by Will Sijp, the supervisor of this project, whose detailed analysis can be found in the references. What follows is a short recap.
PCA is a statistical method that reduces complex datasets into a smaller number of uncorrelated components, each capturing a dominant source of variation. In our project, the PC1 explains 96% of the variance, while PC2 and PC3 account for 2.5% and 0.7%, respectively. Together, these three components capture nearly all the variability in the housing indexes.
The loadings from the PCA highlight which regions contribute most strongly to each component. For example, PC2, labelled the “mining” factor, shows high positive loadings in resource-dependent regions, while service-oriented regions display negative loadings. This pattern suggests that PC2 captures the influence of the mining sector on housing markets. By contrast, PC1, the “market” factor, has positive loadings across nearly all regions, reflecting the common national trend in housing prices.
Factor-Based Substitutes for Principal Components
After defining the factors associated with each principal component (PC), we introduce a linear regression that combines selected housing indexes to approximate the PCs, which are referred to as factor proxies. The objective is to identify interpretable city or regional combinations that closely mirror the PCA-derived components. A proxy is considered valid if it achieves a correlation of at least 96% with the corresponding PC, providing statistical evidence of its suitability. This procedure is not strictly systematic but rather heuristic, involving an iterative search for the most representative mix of cities or regions. More specifically:
For PC1 proxy the national spread (U), the factor model is specified as: \(\mu_r = \beta_r U + \varepsilon\)
For the PC2 proxy, the mining spread (Perth–Sydney spread, \(\delta_{PS}\)), the factor model is specified as: \(\mu_r = \beta_r U + \lambda_r \delta_{PS} + \varepsilon\)
For PC3 proxy the lifestyle spread (WIP)
Forecasting of Factors
Once the factor proxies were defined, we modelled each factor series separately using time series forecasting methods. The process included exploratory analysis, stationarity checks, and STL decomposition to characterise the data. Guided by these diagnostics, we selected suitable ARIMA-type models, enhancing them with Fourier terms to capture seasonality and intervention dummies to address structural breaks.
Integration into a Factor Modelling Framework
The linear model regresses each regional housing index on the factor proxies, yielding coefficients that represent the sensitivity of local markets to the underlying factors: \(\mu_r = \beta_{National} + \lambda_r \delta_{Mining} + \gamma_r \delta_{lifestyle} + \varepsilon\), the coefficients β, λ and γ are estimated for each region r. These coefficients help to capture the response of the local market to each factor. Model performance was evaluated using R-squared and residual diagnostics to check that key assumptions held. Forecasts of the factor proxies were then substituted into the model to obtain projected housing indexes. To account for changing dynamics, coefficients were re-estimated with rolling windows, allowing them to vary over time and better capture evolving market conditions.
Factor Model Enhancement
Residual diagnostics from the static factor model indicated autocorrelation, suggesting that regional price dynamics were not fully captured by contemporaneous factors alone. To address this, the model was extended using an ARIMAX framework, retaining the same factor proxies as exogenous variables while incorporating autoregressive and moving-average terms to account for temporal dependence. An expanding-window approach was adopted to re-estimate models as new data became available, enabling coefficients to adapt to evolving market conditions and improving both model stability and forecast accuracy.
Use Case
The resulting model provides a practical framework for analysing and forecasting regional housing markets across Australia. By quantifying how each region responds to national, mining, and lifestyle factors, the model enables identification of regions most exposed to structural trends, such as shifts in mining investment or lifestyle migration. These outputs can inform policy analysis, highlighting areas vulnerable to affordability pressures or cyclical volatility, and support investment decisions by revealing which regions are likely to outperform or lag under different macroeconomic conditions.
Datasets
The datasets used in this project come primarily from the Australian Bureau of Statistics (ABS), covering housing price indexes and selected macroeconomic indicators. In addition, we make use of internally developed datasets, including preliminary outputs such as PCA results and estimated static coefficients. All datasets are organised and available in the data folder:
- Housing indexes (log transformed) on both city level and SA4 level,
named
city_indexesandindexes_city_and_sa4 - PCA results including PC scores and loadings, named
df_pcs,df_eofs_city, anddf_eofs_sa4 - Factor proxies identified from PCA results, named
df_factor_trends - Regression coefficients estimated from the factor proxies, named
df_reg_coefs - Geographic data for SA4 regions, named
SA4_2021_AUST_SHP_GDA2020.shp - Economic data including GDP, exchange rate, and mining expenditure,
named
data.xlsx,key_economic_data.xlsx, andmining_expenditure.xlsx(WIP)
Project Limitation
This project faces several dataset-related limitations, including but not limited to the following:
- Historical Scope: PCA results and coefficients are based on 25 years of data (1998–2023), which may not fully reflect future patterns.
- Measurement Bias: Indexes rely on hedonic and repeat-sales methods, which involve assumptions that may introduce bias.
- Proxy Validity: Factor proxies are heuristic choices and may not perfectly represent the underlying principal components.
- Data Sensitivity: Outcomes are highly data-driven and depend strongly on the chosen datasets and methods.
- Structural Shocks: Events like the mining boom, COVID-19, or policy shifts may not be well captured by static models or PCA.
From Principal Components to Factor Proxies
As noted earlier, the PCA component of this work was conducted by Will Sijp as part of Sijp et al. (2025), and this project builds upon that foundation by constructing factor trends to serve as proxies for the principal components. The criterion for a good proxy is achieving a statistical correlation of around 96% with the original PC series.
To develop these proxies, we:
- Analysed the PCA loadings to identify regions with the strongest positive and negative contributions to each component.
- Formulated candidate proxies as linear combinations of housing indexes from these influential regions, focusing on those with clear economic or interpretive relevance.
- Assessed the strength of each candidate by measuring its correlation with the corresponding principal component score.
To illustrate this process, we compared two candidate combinations for representing PC2 (the mining factor): the Brisbane–Sydney and Perth–Sydney spreads. By examining the shapes of the standardised series and testing their correlations with PC2, we found that the Perth–Sydney spread aligns more closely with the original component.
Its scatterplot shows a tighter fit along the regression line, and the correlation with PC2 is marginally higher (98% versus 96% for the Brisbane–Sydney spread). Although the difference is small and not statistically significant, the Perth–Sydney spread provides the more robust proxy for PC2.
Exploratory Analysis
The exploratory analysis utilises visualisations to characterise the dynamics of the national trend (U) and the Perth–Sydney spread (δPS), thereby clarifying their underlying properties.
Time-series plots:
U rises steadily from 2003 to 2012, transitions into a slower growth phase until 2019, then undergoes a sharp regime shift during 2020–2022 (the COVID period) followed by persistently elevated and volatile levels, with growth rates remaining time-varying throughout.
δPS displays a prolonged, uneven rise from 2003 to 2012, followed by a sharp regime shift, then COVID-related fluctuations, with the level adjusting and showing time-varying patterns.
Seasonal plots:
U has apparent within-year slopes being dominated by the strong upward trend; no systematic month-of-year effects.
δPS shows little evidence of month-to-month seasonality, with no particular month standing out, though its overall level varies between years.
Seasonal sub-series plot:
Each monthly panel of U shows a steady upward rise with mean value across months remaining similar, reinforcing that the series is driven more by trend than by seasonality.
The δPS seasonal sub-series plot reinforces the lack of a dominant seasonal pattern, as the monthly profiles are similar in shape and the corresponding means remain relatively flat and close to zero.
Forecasting with the Factor Model
With the factors now established, we proceed to fit time series forecasting models to each series individually. As a first step, we examine the series in more detail. STL decomposition plots allow us to closely inspect the trend, seasonality, and remainder components, which helps determine the most suitable forecasting models for the series’ characteristics.
Stationary tests (ADF and KPSS) indicate that both the national index (U) and the Perth–Sydney spread (PS) are non-stationary, so differencing is required. However, based on economic intuition that the PS series may be mean-reverting, we applied the Zivot–Andrews (ur.za) test. This confirmed that PS could be stationary with several structural break.
The dataset was partitioned into training and testing subsets, with one year of observations allocated to the testing set. Given that the data are recorded at a monthly frequency and span only 25 years, this allocation represents an optimal balance between ensuring sufficient data for model training while preserving an adequate sample for evaluation.
National Trend
The STL decomposition of the national trend U reveals a smooth and persistent upward trajectory in the trend component, while the seasonal component is of relatively minor magnitude, however the annual seasonality is strong and repeating. To stabilise variation in the seasonal component, a Box–Cox transformation was applied using the Guerrero method, which produced a λ value of 0.9. Given its proximity to one, the transformation had a negligible effect on the modelling outcome. The remainder component displays mostly small noise, with some bigger shocks.
The modelling strategy was guided by exploratory data analysis and
the STL decomposition, which suggested a random walk with drift,
ARIMA(U ∼ pdq(0,1,0)), as an appropriate baseline
specification.
To refine the model, we subsequently evaluated neighbouring specifications with alternative autoregressive and moving average terms, using AICc values and residual diagnostics as selection criteria.
In the modelling process, we evaluated a range of specifications
informed both by diagnostic checks and prior knowledge. Inspection of
the residuals suggested some semi-annual variation, and Fourier terms
with K = 2 were added to account for potential
higher-frequency seasonality. We also investigated seasonal extensions
with P = 1, alongside piecewise specifications with knots
aligned to major economic events: the Global Financial Crisis (2008),
the end of the mining investment boom (2012), and the onset of the
COVID-19 pandemic (2020). These variations were assessed with respect to
residual behaviour, statistical significance, and their ability to
capture shocks apparent in the remainder component.
Among the fitted models, two candidates emerged as particularly strong. The first, an ARIMA(2,1,1) with Fourier terms (K = 2) and drift. A second competitive specification combined Fourier terms (K = 2) with intervention dummies (step and pulse) and an ARIMA(2,1,1) structure.
## # A tibble: 8 × 8
## .model sigma2 log_lik AIC AICc BIC ar_roots ma_roots
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <list> <list>
## 1 step 0.00000672 1338. -2650. -2649. -2602. <cpl [26]> <cpl [1]>
## 2 arima211 0.00000700 1331. -2640. -2639. -2599. <cpl [26]> <cpl [1]>
## 3 arima110 0.00000712 1327. -2637. -2636. -2604. <cpl [25]> <cpl [0]>
## 4 arima111 0.00000715 1327. -2635. -2634. -2598. <cpl [25]> <cpl [1]>
## 5 sarima211 0.00000785 1312. -2611. -2611. -2589. <cpl [14]> <cpl [1]>
## 6 autoarima 0.00000774 1308. -2609. -2609. -2598. <cpl [24]> <cpl [0]>
## 7 piece 0.00000791 1312. -2606. -2606. -2573. <cpl [1]> <cpl [25]>
## 8 rw_drift 0.0000416 1066. -2127. -2127. -2116. <cpl [0]> <cpl [12]>
Although both models fall short of strictly meeting the residual whiteness criterion, particularly since the degrees of freedom were not clearly defined in the test results, we use them here mainly for relative comparison of performance. Visual inspection of the residual plots suggests that the residuals are approximately normally distributed, with only a small share of autocorrelation coefficients exceeding the significance bounds in the ACF plot. As a general rule of thumb, if fewer than about 5% of the lags lie outside these bounds, the residuals can reasonably be treated as white noise.
## # A tibble: 8 × 3
## .model lb_stat lb_pvalue
## <chr> <dbl> <dbl>
## 1 arima211 18.4 0.103
## 2 step 20.4 0.0595
## 3 sarima211 21.7 0.0411
## 4 autoarima 23.0 0.0279
## 5 arima110 24.7 0.0165
## 6 arima111 24.8 0.0157
## 7 piece 28.9 0.00411
## 8 rw_drift 885. 0Mining Trend
The mining trend (PS) oscillates around zero and appears mean-reverting with no evidence ofpermanent trend. Seasonality is present but relatively stable across years, while longer multi-year swings suggest the need for more flexible seasonal modelling. The residuals still display structure, with a clear AR(1) cut-off rather than pure white noise.
The mining trend (PS) captures the dynamics of the resource boom, which we interpret as a one-off structural break rather than a permanent driver. Prices surged sharply during the 2000s mining boom, collapsed around 2012, and have remained stagnant since. If this raw factor is carried forward into forecasts, the boom–bust cycle dominates and produces unstable predictions with excessively wide intervals.
To address this, we treat the boom as a temporary structural shock. We introduce a dummy variable (set to 1 during the boom/decline period and 0 otherwise) when modelling PS, and then forecast only the residual stationary component. This approach effectively assumes that another boom of comparable scale is unlikely.
The boom and decline phases were identified using the
breakpoints function from the strucchange
package, which detected three structural breaks in the series. The
segment containing the global peak was classified as the “boom” period,
followed by the “decline” period. These periods were then encoded into
dummy variables for use in the modelling process.
To model the series with the dummy variables, essentially we fit an ARIMA model to the residuals after accounting for the boom and decline periods. The final selected model was an ARIMA(2,0,1) with the boom and decline dummies included as regressors.
## Series: PS
## Model: LM w/ ARIMA(2,0,1) errors
##
## Coefficients:
## ar1 ar2 ma1 boom decline
## 1.8741 -0.8763 -0.3764 -0.0081 0.0014
## s.e. 0.0389 0.0388 0.0829 0.0041 0.0041
##
## sigma^2 estimated as 4.359e-05: log likelihood=1061.23
## AIC=-2110.45 AICc=-2110.16 BIC=-2088.33
Forecast Performance
For the forecasting stage, we use the best-performing models (ARIMA(2,1,1) with Fourier terms (K = 2) for U and ARIMA(2,0,1) for PS) identified earlier to project values 10 years ahead (120 months). We generate forecasts with prediction intervals at both 80% and 95%. These intervals show the range in which future values are likely to fall, giving us a measure of uncertainty, an 80% interval means there is an 8-in-10 chance that the true value will lie within that range.
For evaluation, we set aside one year of data (2023) as a test set. Although a typical split is about 20% of the dataset, our series only spans 25 years, so using 20% would cut off too much valuable information for forecasting. A one-year test period strikes a better balance.
The forecasts indicate that the national trend is expected to rise, with an 80% prediction interval width of 0.6845, reflecting higher uncertainty. In contrast, the mining trend (PS) shows a flatter path, with an 80% interval width of 0.4538. This narrower range reflects greater stability, largely because we excluded the one-off boom and decline period, which would otherwise make the series more volatile.
The test set results show that the chosen model for U performs consistently with very low error (RMSE = 0.0076, negligible relative to the scale of U). For PS, the model incorporating boom/decline dummies produces out-of-sample forecasts that are on average about 7% off, which is acceptable given the series’ inherently difficult-to-predict behaviour (mining cycle is more volatile than the national market). While the ACF1 values indicate some remaining autocorrelation in the residuals, overall both models demonstrate solid performance.
## # A tibble: 2 × 6
## series .model .type RMSE MAE MAPE
## <chr> <chr> <chr> <dbl> <dbl> <dbl>
## 1 U (national) arima211 Test 0.00756 0.00487 0.261
## 2 PS (mining) arima201 Test 0.0116 0.00807 6.83Factor Model Application
Goodness of Fit
We now turn to evaluating how well the factor model captures the variation in individual major city and SA4 indexes. R² measures the share of variation in each major city or SA4 housing index that our factor model explains.
For major cities and rest-of-state: Most city and regions have an R² value above 0.98. The extremes highlight clear spatial patterns:
Top cities and rest-of-state: Large east-coast capitals and their surrounding belts.
Bottom cities and rest-of-state: Smaller or resource-dependent markets.
| Rank | Region | R² | Rank | Region | R² |
|---|---|---|---|---|---|
| 1 | GREATER MELBOURNE | 0.9985 | 15 | REST OF WA | 0.9697 |
| 2 | REST OF NSW | 0.9985 | 14 | GREATER PERTH | 0.9735 |
| 3 | REST OF QLD | 0.9983 | 13 | REST OF NT | 0.9800 |
| 4 | GREATER SYDNEY | 0.9981 | 12 | GREATER HOBART | 0.9857 |
| 5 | GREATER BRISBANE | 0.9979 | 11 | GREATER DARWIN | 0.9860 |
For SA4 regions: The SA4 regions shares the same stories. R² is tightly concentrated near 1.0 with the majority of SA4s exceed 0.98, with only a small left tail (a few around 0.92 – 0.95). The top and bottom lists identify the extremes and show recurring patterns across those regions:
- Top 10 regions: Concentrated in the large east-coast capitals and surrounding belts.
- Bottom 10 regions : Dominated by Western Australia and Queensland’s mining exposed regions.
| Rank | Region | R² | Rank | Region | R² |
|---|---|---|---|---|---|
| 1 | MELBOURNE - SOUTH EAST | 0.9982 | 87 | WESTERN AUSTRALIA - OUTBACK (NORTH) | 0.9184 |
| 2 | MELBOURNE - OUTER EAST | 0.9979 | 86 | MANDURAH | 0.9460 |
| 3 | BRISBANE - WEST | 0.9975 | 85 | WESTERN AUSTRALIA - OUTBACK (SOUTH) | 0.9601 |
| 4 | BRISBANE - SOUTH | 0.9974 | 84 | BUNBURY | 0.9606 |
| 5 | BRISBANE - NORTH | 0.9974 | 83 | WESTERN AUSTRALIA - WHEAT BELT | 0.9635 |
| 6 | ADELAIDE - WEST | 0.9972 | 82 | MACKAY - ISAAC - WHITSUNDAY | 0.9678 |
| 7 | MELBOURNE - NORTH EAST | 0.9972 | 81 | PERTH - SOUTH EAST | 0.9691 |
| 8 | ADELAIDE - CENTRAL AND HILLS | 0.9972 | 80 | PERTH - NORTH EAST | 0.9695 |
| 9 | SYDNEY - BAULKHAM HILLS AND HAWKESBURY | 0.9970 | 79 | PERTH - SOUTH WEST | 0.9732 |
| 10 | WIDE BAY | 0.9970 | 78 | CENTRAL QUEENSLAND | 0.9743 |
Both SA4 regions and major cities display highly concentrated R² distributions with major cities showing less variation.
The choropleth reinforces the findings from the summary table, as we can see that the east-coast capitals and coastal belts from Melbourne through Sydney into the South-East of Queensland show the strongest fit, with most regional NSW/VIC also high. While fit weakens inland and to the west, notably across Western Australia and along Queensland’s mining belts; The dark purple areas mark the lowest R² and are typically remote and resource-exposed regions with possible boom–bust timing and sparse transactions that our three factors don’t fully capture.
Hence, we’ll prioritise Western Australia and Queensland’s mining-exposed regions for residual diagnostics, where we expect autocorrelation and regime shifts for trial enhancements.
Residual Autocorrelation
At this stage, the model does not incorporate any time dynamics in the residuals. While it explains a substantial share of the variance for both cities and regions, the trending and autocorrelated nature of housing indexes means that the residuals themselves display extremely strong positive autocorrelation. This is evident in the Durbin–Watson statistics, which are close to zero across all regions: Major cities cluster around 0.02 to 0.05, while SA4 regions have a slightly broader range, extending up to 0.2, but still remain far below the benchmark value of 2. This pattern underscores the limitations of using a static regression framework without accounting for time-series dynamics.
## # A tibble: 2 × 6
## region_level dw_min dw_max dw_mean dw_median range
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 major_city 0.0146 0.104 0.0338 0.0282 0.0891
## 2 sa4_name 0.0211 0.201 0.0680 0.0598 0.180
Coefficient Pattern
While R² and Durbin–Watson assess overall fit and residual behaviour, they do not show how regions load onto each factor. Examining coefficient patterns adds this context, revealing whether markets move uniformly with the national trend or diverge due to mining influences.
The distribution of market coefficients is tightly centred around 1.0 with relatively little variation. This indicates that most regions respond in a similar way to the national housing trend. In contrast, the mining coefficients are far more dispersed with a long right tail and several extreme outliers, suggesting that exposure to mining-related dynamics differs substantially across regions.
The scatterplot analysis reveals a clear negative relationship between the market and mining coefficients. Regions with a strong loading on the national market factor tend to exhibit weaker or even negative loadings on the mining factor and vice versa.
Model Enhancement
From the analysis of the OLS factor model, several limitations were identified:
Strong residual autocorrelation, reflecting the trending and persistent nature of housing price movements.
Spatial variation in model fit, with mining-exposed regions showing weaker R² values and more dispersed coefficient estimates.
Lack of temporal adaptability, as static coefficients cannot capture evolving relationships between regional markets and national or sectoral factors.
To address these issues, two key enhancements were introduced:
ARIMAX-based residual control, incorporating autoregressive and moving-average terms to explicitly model autocorrelation and stabilise residuals.
Time-varying estimation using expanding windows, allowing coefficients to evolve with changing market conditions and improving responsiveness to structural shifts in housing dynamics.
Model Enhancement I: ARIMAX-Based Residual Control
The ARIMAX approach retains fixed regression coefficients while modelling the residuals with an ARIMA process. Several baseline specifications (1,1,0),(0,1,1),(0,1,0),(1,1,1) were tested, and the optimal model was chosen by balancing AICc performance with Ljung–Box test results, ensuring that the residuals approximated white noise.
The ARIMA(1,1,0) specification achieved the strongest overall performance, with 69.3% of SA4 regions exhibiting residuals whose Ljung–Box p-values exceeded the 5% threshold, indicating adequately whitened residuals.
For the remaining regions that did not meet this criterion, we analysed their residual patterns to identify the underlying causes of autocorrelation. Regions displaying clear seasonal patterns were refitted using a seasonal ARIMA(1,0,0)[12]model, while those showing short-lag dependence were assigned an AR(2) structure. Incorporating these targeted refinements raised the proportion of regions passing the diagnostic test to 86.4%, substantially improving overall model adequacy.
(need to add ARIMAX models explanation here)
| pass | mean_AIC | model |
|---|---|---|
| 0.6931818 | -1923.782 | Baseline ARIMA(1,1,0) |
| 0.1000000 | -1947.994 | Short lag ARIMA(2,1,0) |
| 0.0625000 | -1873.096 | Seasonal ARIMA(1,1,0)(1,0,0)[12] |
This section compares the coefficient distributions for three factors across two modelling approaches: OLS and ARIMAX. The OLS model (blue) represents a static regression using contemporaneous factors only, while the ARIMAX model (red) incorporates time-series autocorrelation to account for temporal dependencies in the residuals.
Market Factor: Both models yield strong positive coefficients with tightly clustered distributions. The medians are nearly identical, and the IQRs are narrow. This consistency confirms that national market trends exert a robust and uniform influence on regional price movements, irrespective of the modelling framework.
Mining Factor: Median coefficients are close to zero in both models. However, the OLS model exhibits a wider spread, including more positive outliers. In contrast, ARIMAX produces smaller and more stable estimates. This suggests that once time-series persistence is accounted for, the mining signal is partially absorbed, reducing its apparent impact while retaining its relevance.
Lifestyle Factor: Both models show similar distributions, with medians around 0.3 and comparable IQRs. ARIMAX coefficients are more tightly clustered, indicating a slight attenuation of the lifestyle signal when autocorrelation is modelled. This implies that lifestyle effects are modest and less regionally variable once temporal dynamics are considered.
Model Enhancement II: Time-dependent Coefficients
(WIP - rolling ridge regression) Forecasts for each major city and SA4 region are obtained by combining the projected factor trends with the static coefficients from the factor model. Specifically, the estimated loadings are applied to the forecasted values of factors to produce region-specific housing price index projections.
As the coefficients are fixed, the projected indexes evolve in parallel with the factor trends, scaled by their respective loadings, as illustrated in the accompanying figure. To capture greater regional heterogeneity, the subsequent stage of the analysis will extend the model to incorporate time-varying coefficients.
Rolling ridge regression is employed as small and noisy samples (particularly with volatile factors such as mining and lifestyle, can lead to instability in OLS-based rolling coefficient estimates) which adds a penalty term (λ) to the loss function to shrink coefficient estimates towards zero. This regularization helps reduce variance and improve out-of-sample performance, especially when predictors are correlated or the sample size is limited.
We set a 60-month window to captures regime shifts faster and used grid search to identify the global λ that minimises out-of-sample RMSE across all regions. This λ is then applied in a rolling regression framework to estimate time-varying regional coefficients. The optimal λ, 0.0475, indicates a light penalty that nonetheless helps stabilise the estimates. After that we fit rolling ridge regressions for each region using the chosen λ. The resulting time-varying coefficients capture evolving regional sensitivities to the national, mining and lifestyle trends.
We apply a rolling regression using a fixed 5-year window. For each region, the data are split into overlapping windows of equal length, each shifted forward by one month. Within each window we fit a penalised linear regression to estimate the intercept and factor coefficients. The resulting coefficients are recorded at the end of the window, then the window is advanced and the process repeats. This produces a time-series of coefficients for every region showing how the influence of the underlying factors evolves gradually over time.
(NEW - expending window)
Assessing the Boundedness of Market-Driven Forecasts
To evaluate the reliability and stability of the regional housing forecasts, we assessed how bounded the projections are when driven solely by the national market factor.
The analysis was conducted using coefficients derived from both the OLS and ARIMAX specifications. While the OLS model provides a static benchmark, the ARIMAX model extends it by accounting for temporal dependence and reducing residual autocorrelation. In both cases, the mining factor spread was treated as the only varying source of uncertainty, with all other factors held constant. For each region, the 10-year growth implied by the market factor was projected, and a 95 per cent multiplicative uncertainty band was computed based on the volatility of the mining factor and the residual variance from each model.
| bracket | n_regions | avg_error_pct | min_error_pct | max_error_pct |
|---|---|---|---|---|
| 0–10% | 59 | 3.89 | 1.18 | 9.53 |
| 10–20% | 24 | 15.32 | 10.35 | 19.38 |
| 20%+ | 5 | 23.34 | 21.08 | 25.31 |
